This invention pertains to microlithography (transfer of a pattern, defined by a reticle or mask, to a xe2x80x9csensitivexe2x80x9d substrate) using a charged particle beam (e.g, electron beam or ion beam). Microlithography is a key technology used in the manufacture of microelectronic devices such as integrated circuits, displays, thin-film magnetic pickup heads, and micromachines. More specifically, the invention is directed to reducing proximity effects as manifest on the pattern as transferred to the substrate. Even more specifically, the invention pertains to methods for calculating exposure dose at specified regions of the sensitive substrate so as to determine expected respective proximity effects at the specified regions. The invention also pertains to methods for fabricating a reticle, taking into account the results of the proximity effect determinations, that produces less proximity effects during transfer of the reticle pattern to the substrate.
Essentially all contemporary methods for fabricating microelectronic devices involve microlithography steps. In the microlithography step, a pattern defined on a reticle or mask is transferred to a xe2x80x9csensitivexe2x80x9d substrate such as a semiconductor wafer or the like. The devices are formed on the substrate as respective xe2x80x9cchipsxe2x80x9d that are separated later from each other by xe2x80x9cdicingxe2x80x9d the wafer. xe2x80x9cSensitivexe2x80x9d means that the substrate is coated with a substance, termed a xe2x80x9cresist,xe2x80x9d that can be imprinted with a pattern exposed onto the resist using an energy beam. Exemplary energy beams used for microlithography include light, X-rays, and charged particle beams.
A typical pattern includes a large number of pattern elements or features. As the pattern is exposed onto the substrate, the pattern elements are formed by differential exposure of the substrate, i.e., certain areas of the resist receive a relatively high exposure dose and other areas receive a relatively low exposure dose. After exposure, areas of resist where the exposure dose (cumulative exposure-irradiation energy) exceeds a threshold value are removed (in the case of a positive resist) or left on the substrate (in the case of a negative resist) by developing the resist. To form a pattern element having a respective desired shape profile on the sensitive substrate, it is necessary to calculate whether the exposure dose at the location on the substrate where the pattern element is exposed is higher than a specified threshold. It also is necessary to configure the pattern element on the reticle such that portions of the pattern element corresponding to areas in which the localized exposure dose on the substrate exceeds the threshold nevertheless form the pattern element with the desired shape profile on the substrate.
In charged-particle-beam (CPB) microlithography, proximity effects arise under conditions in which actual localized exposure doses (e.g., exposure doses at single pattern elements) vary according to the nearness, respective profiles, and distribution of neighboring pattern elements, due to scattering of electrons in the resist and from the substrate. More specifically, proximity effects arise due to the scattering of charged particles, incident upon the surface of the sensitive substrate, at small angles that reduce the exposure dose at specified locations. These small-angle scattering events are termed xe2x80x9cforward-scattering.xe2x80x9d Proximity effects also arise due to the scattering of charged particles at wide angles that contribute exposure energy to neighboring unexposed areas. These wide-angle scattering events are termed xe2x80x9cback-scattering.xe2x80x9d Whenever a proximity effect occurs, the exposure dose at a respective location on the sensitive substrate differs from what is expected or desired at the location. As a result, the pattern element that is formed at the location on the substrate usually has a profile that undesirably is different from the desired profile.
Conventional methods for reducing proximity effects generally involve making localized exposure doses closer to desired respective doses. For example, certain methods involve changing and adjusting localized exposure doses by changing beam intensity (dose modulation); others involve changing the profiles of pattern elements as defined on the reticle (xe2x80x9clocal resizingxe2x80x9d).
On the substrate, a planar distribution of exposure dose in the resist from irradiation, by a charged particle beam, of a point (x, y) on the surface of the resist can be expressed as the sum of a Gaussian distribution (i.e., a double Gaussian distribution):       E    ⁢          (              x        ,        y            )        =                    (                  1                      1            +            η                          )            ⁢              (                  1                      πσ            f            2                          )            ⁢              exp        ⁡                  [                      -                                          (                                                      x                    2                                    +                                      y                    2                                                  )                                            σ                f                2                                              ]                      +                  (                  η                      1            +            η                          )            ⁢              (                  1                      πσ            b            2                          )            ⁢              exp        ⁡                  [                      -                                          (                                                      x                    2                                    +                                      y                    2                                                  )                                            σ                b                2                                              ]                    
The standard-deviation terms, "sgr"f and "sgr"b, are broadening terms known as the xe2x80x9cforward-scattering diameterxe2x80x9d and xe2x80x9cback-scattering diameter,xe2x80x9d respectively; and xcex7 is a ratio of exposure energy from back-scatter to exposure energy from forward-scatter (i.e., the xe2x80x9cback-scatter fractionxe2x80x9d). If defocusing in the projection-optical system of the microlithography apparatus is taken into consideration, then the sum of squares of the magnitude of defocusing and the forward-scattering diameter is calculated and substituted as a new "sgr"f.
The following discussion refers to mathematical expressions based on back-scattering diameter "sgr"b. Expressions based on forward-scattering diameter "sgr"f or optical-system defocusing can be set forth in a similar manner, wherein the back-scattering term is substituted with the forward-scattering term or optical-system-defocusing term.
If a pattern of reference figures (reference elements) is configured as N rectangles each having diagonal apices at the coordinates (x1j, y1j), (x2j, y2j) (where j=1, 2, 3, . . . , N), then the back-scattering energy Eb(x, y) at a location (x, y) can be expressed by integrating the E(x, y) expression above, yielding the following:             E      b        ⁡          (              x        ,        y            )        =            ∑      j        ⁢                  [                              erf            ⁡                          (                                                (                                      x                    -                                          x                                              1                        ⁢                        j                                                                              )                                                  σ                  b                                            )                                -                      erf            ⁡                          (                                                (                                      x                    -                                          x                                              2                        ⁢                        j                                                                              )                                                  σ                  b                                            )                                      ]            xc3x97              [                  xe2x80x83                ⁢                              erf            ⁡                          (                                                (                                      y                    -                                          y                                              1                        ⁢                        j                                                                              )                                                  σ                  b                                            )                                -                      erf            ⁡                          (                                                (                                      y                    -                                          y                                              2                        ⁢                        j                                                                              )                                                  σ                  b                                            )                                      ]            
wherein xe2x80x9cerfxe2x80x9d denotes an error function.
This calculation yields a sum corresponding only to the specific number of reference figures to which reference is being made. Consequently, a problem with this calculation is that, as the number N of reference figures increases (with an increase in the density and/or complexity of circuit elements in the pattern), the calculation time increases commensurately.
A conventional method for addressing this problem involves the use of xe2x80x9crepresentative figures,xe2x80x9d as exemplified in FIGS. 9(A)-9(B). In FIG. 9(A), a pattern region 91 is depicted containing multiple reference figure 93. The region 91 is divided (along dashed lines) into multiple sub-regions 92. Each sub-region 92 is small relative to the back-scatter diameter "sgr"b and serves as the fundamental unit of pattern area on which calculations of local exposure dose are based. By performing exposure-dose calculations based on the contents of specified sub-regions, the number of reference figures used for calculating back-scatter energy is reduced (with a concomitant reduction in calculation time) compared to making calculations based on each individual actual pattern element of the pattern. In FIG. 9(B), a single respective representative figure 94 is derived for each sub-region 92. Each representative figure 94 has the same total surface area and centroid as the respective reference figure 93 in the respective sub-region.
However, in certain instances (e.g., with a pattern for a LSI device) an actual pattern element 103 can have a marked dimensional bias. For example, FIG. 10 depicts a region 102 of a pattern portion 101 (in FIG. 10, the dashed lines denote respective coordinate axes). As shown in the upper portion of FIG. 10, the pattern element 103 extends across the upper portion of the region 102. The pattern element 103 has a horizontally extended, narrow rectangular profile, with a large difference in its length versus its width. A square representative figure 104, as determined using conventional methods, for the pattern element 103 is shown in the lower portion of FIG. 10. With respect to the pattern element 103, if the square representative figure 104 were used for calculating local exposure dose, then the result of the calculation would be quite inaccurate for the particular pattern element 103. On the other hand, if the representative figure 104 could be made horizontally rectangular in profile, then the representative figure would more appropriately represent the pattern element 103 for purposes of calculating exposure dose. However, to represent one or more pattern elements using a rectangular representative figure, it is necessary to determine the appropriate aspect ratio for the rectangle. No currently known methods are available for determining suitable respective aspect ratios for rectangular representative figures or for configuring rectangular representative figures of appropriate respective aspect ratios.
Turning now to FIG. 11, consider a region 110 of a pattern portion 109. The region 110 contains four pattern elements. (The dashed lines denote respective coordinate axes.) In a first example (shown at the top of FIG. 11), the pattern elements 111 are clustered in the center of the region 110. In a second example (shown in the middle of FIG. 11), the pattern elements 112 are separated more distantly from the center of the region 110. If a single representative figure 113 is derived for all four elements 111 or 112 using conventional methods, then the representative figure 113 would be the same in both instances, as shown at the bottom of FIG. 11. In both instances, the representative figure 113 has the same total area and centroid as the group of pattern elements 111 or 112 it represents. Nevertheless, especially for the second example involving the pattern elements 112, the fidelity with which the representative figure 113 represents the actual pattern elements 112 can be poor, which causes substantial inaccuracy in the results of the exposure-dose calculations.
In view of the shortcomings of conventional methods as summarized above, an object of the invention is to provide methods for accurately calculating exposure-dose (exposure irradiation energy) at specific locations on a sensitive substrate. The methods include determining representative figures in regions of the pattern, wherein each representative figure more accurately reflects the aspect ratio(s) of the respective pattern element(s) in the region and more accurately reflects the distribution of the actual pattern element(s) in the region. The results of these determinations are used in calculations to determine proximity effects, to configure pattern elements as defined on the reticle so as to reduce the proximity effects, and to operate the charged-particle-beam (CPB) microlithography apparatus such that exposure-irradiation doses are adjusted as required to reduce proximity effects.
To such ends, and according to a first aspect of the invention, methods are provided for quantifying an exposure dose received in a region on a surface of the sensitive substrate exposed with a pattern using a charged particle beam. The methods are set forth in the context of a charged-particle-beam microlithography method in which a pattern, defined by a reticle extending in an X-Y plane that is perperpendicular to a Z-axis serving as a microlithographic optical axis, is transferred by a charged particle beam to a sensitive substrate. According to the method, the pattern is divided into multiple pattern regions. (Not all of the region need contain pattern elements.) In a pattern region containing at least one pattern element or portion of a pattern element, respective values of multiple parameters are determined concerning the pattern element(s) and/or portion(s) thereof in the region. (Herein, the term xe2x80x9cpattern elementxe2x80x9d in the context of what is contained within a region can be a respective portion of a pattern element that extends into multiple regions.) The parameters include centroid position, total surface area, and moment of inertia relative to an axis passing through the centroid. For the pattern element(s) in the pattern region, a respective representative figure is calculated having the same values of the characteristic parameters. Using the representative figure instead of the pattern element(s) in the pattern region, the exposure dose (E(x, y)) in the corresponding region on the surface of the substrate is calculated.
More specifically, for regions that contain at least one pattern element or portion of a pattern element, the parameters for the pattern element(s) in the pattern region are the total surface area (S), coordinates (Gx, Gy) of the centroid, sum (Ix) of the moments of inertia of the pattern element(s) relative to an axis parallel to the X-axis and passing through the centroid (Gx, Gy), and sum (Iy) of the moments of inertia of the pattern element(s) relative to an axis parallel to the Y-axis and passing through the centroid (Gx, Gy). The representative figure has a rectangular profile with diagonal corners at points (rx1, ry1), (rx2, ry2); wherein:       u    =                  (                  S                                                    I                x                                      ⁢                                          I                y                                                    )                    1        2                        r      x1        =                  G        x            -                        1          2                ⁢                              I            y                          ⁢        u                        r      x2        =                  G        x            +                        1          2                ⁢                              I            y                          ⁢        u                        r      y1        =                  G        y            -                        1          2                ⁢                              I            x                          ⁢        u                        r      y2        =                  G        y            +                        1          2                ⁢                              I            x                          ⁢        u            
The pattern region can contain a number (j) of respective pattern elements, wherein jxe2x89xa71 and Sj is the area of each respective pattern element. In this instance, the centroid of the pattern element(s), regarded collectively in the region, has coordinates Gxj, Gyj, each moment of inertia around the axis x=Gx is denoted Ixj, and each moment of inertia around an axis y=Gy is denoted Iyj. Under such conditions,                     S        =                              ∑            j                    ⁢                      S            j                                              xe2x80x83                                          G          x                =                              ∑            j                    ⁢                                    G              xj                        ⁢                                          S                j                            /              S                                                                    G          y                =                              ∑            j                    ⁢                                    G              yj                        ⁢                                          S                j                            /              S                                                                        I          x                =                              ∑            j                    ⁢                      I            xj                                                        I          y                =                              ∑            j                    ⁢                      I            yj                              
In the subject method, a factor (f), associated with respective broadening of the pattern element(s) in the pattern region, can be determined. The irradiation intensity of the charged particle beam on the representative figure is f times the irradiation intensity, and the representative figure is configured to have an area of 1/f. In this instance, the parameters for the pattern element(s) in the pattern region are the total surface area (S), coordinates (Gx, Gy) of the centroid, sum (Ix) of the moments of inertia of the pattern element(s) relative to an axis parallel to the X-axis and passing through the centroid (Gx, Gy), and sum (Iy) of the moments of inertia of the pattern element(s) relative to an axis parallel to the Y-axis and passing through the centroid (Gx, Gy). The representative figure is regarded as having a rectangular profile with diagonal corners at points (Rx1, Ry1), (Rx2, Ry2). The exposure-irradiation energy of the charged particle beam on the representative figure is a multiple f of a true irradiation energy, wherein:
      f    =                  S        2                    12        ⁢                              I            x                          ⁢                              I            y                                    U    =                  2        ⁢                  3                            S                        R      x1        =                  G        x            -                        1          2                ⁢                              I            y                          ⁢        U                        R      x2        =                  G        x            +                        1          2                ⁢                              I            y                          ⁢        U                        R      y1        =                  G        y            -                        1          2                ⁢                              I            x                          ⁢        U                        R      y2        =                  G        y            +                        1          2                ⁢                              I            x                          ⁢        U            
Izxe2x80x2 is a moment of inertia of the representative figure around a line parallel to a Z-axis, perpendicular to the X- and Y-axes, and passing through the centroid (Gx, Gy). Izxe2x80x2 is expressed as follows:                               I          z          xe2x80x2                =                  xe2x80x83                ⁢                                            1              12                        ⁡                          [                                                                    (                                                                  r                        x2                                            -                                              r                        x2                                                              )                                    2                                +                                                      (                                                                  r                        y2                                            -                                              r                        y1                                                              )                                    2                                            ]                                ⁢          S                                        =                  xe2x80x83                ⁢                                            1              12                        ⁡                          [                                                                    (                                                                                            I                          x                                                                    ⁢                      u                                        )                                    2                                +                                                      (                                                                                            I                          y                                                                    ⁢                      u                                        )                                    2                                            ]                                ⁢          S                                        =                  xe2x80x83                ⁢                                            1              12                        ⁡                          [                              (                                                                            I                      y                                        ⁢                                          u                      2                                                        +                                                            I                      x                                        ⁢                                          u                      2                                                                      )                            ]                                ⁢          S                    
The representative figure is determined based on a sum Iz of the individual moments of inertia of the pattern element(s) around a line parallel to the Z-axis and passing through the centroid of the representative figure. The sum Iz is expressed as:       I    z    =                    1        12            ⁡              [                  (                                                    I                y                            ⁢                              U                2                                      +                                          I                x                            ⁢                              U                2                                              )                ]              ⁢    S  
In a situation in which Izxe2x80x2=Iz:                     U        =                                            (                                                I                  z                                                  I                  z                  xe2x80x2                                            )                                      1              2                                ⁢          u                                        =                                            [                                                (                                                            I                      x                                        +                                          I                      y                                                        )                                                                      1                    12                                    *                                      (                                                                  I                        x                                            +                                              I                        y                                                              )                                    ⁢                                      u                    2                                    *                  S                                            ]                                      1              2                                *          u                                        =                              2            ⁢                          3                                            S                              
The representative figure can be configured to have diagonal corners at the points (Rx1, Ry1), (Rx2, Ry2), wherein:       f    =                  S        2                    12        ⁢                              I            x                          ⁢                              I            y                                    U    =                  2        ⁢                  3                            S                        R      x1        =                  G        x            -                        1          2                ⁢                              I            y                          ⁢        U                        R      x2        =                  G        x            +                        1          2                ⁢                              I            y                          ⁢        U                        R      y1        =                  G        y            -                        1          2                ⁢                              I            x                          ⁢        U                        R      y2        =                  G        y            +                        1          2                ⁢                              I            x                          ⁢        U            
The foregoing and additional features and advantages of the invention will be more readily apparent from the following detailed description, which proceeds with reference to the accompanying drawings.